Improved MPS-FE Fluid-Structure Interaction Coupled Method ... · Improved MPS-FE Fluid-Structure...

Copyright © 2014 Tech Science Press CMES, vol.101, no.4, pp.229-247, 2014

Improved MPS-FE Fluid-Structure Interaction CoupledMethod with MPS Polygon Wall Boundary Model

N. Mitsume1, S. Yoshimura1, K. Murotani1 and T. Yamada1

Abstract: The MPS-FE method, which adopts the Finite Element (FE) methodfor structure computation and the Moving Particle Simulation (MPS) method forfluid computation involving free surfaces, was developed to solve fluid-structureinteraction problems with free surfaces. The conventional MPS-FE method, inwhich MPS wall boundary particles and finite elements are overlapped in orderto exchange information at a fluid-structure interface, is not versatile and reducesthe advantages of the software modularity. In this study, we developed a non-overlapping approach in which the interface in the fluid computation correspondsto the interface in the structure computation through an MPS polygon wall model.The accuracy of the improved MPS-FE method was verified by solving a dam breakproblem with an elastic obstacle and comparing the result obtained with that of theconventional MPS-FE method and particle FEM.

Keywords: Fluid-Structure Interaction, Free Surface, Moving Particle Simula-tion Method, Finite Element Method, MPS-FE Method.

1 Introduction

Large facilities such as electric power plants built along coastal regions are vul-nerable to tsunamis. The resulting damage to the equipment and instruments hasthe potential to cause catastrophic harm to people and devastate the locality. TheFukushima Daiichi nuclear disaster in Japan is a prominent example of this. It iseconomically impossible to completely mitigate the effects of disasters of extremeseverity; however, damages and loss can be minimized through appropriate mea-sures.

Regarding tsunami simulation, a great deal of research [Westerink, Luettich, Feyen,Atkinson, Dawson, Roberts, Powell, Dunion, Kubatko, and Pourtaheri (2008); Cre-spo, Gómez-Gesteira, and Dalrymple (2007)] has been conducted on the simulation

1 The University of Tokyo, Hongo, Bunkyo, Tokyo, Japan.

230 Copyright © 2014 Tech Science Press CMES, vol.101, no.4, pp.229-247, 2014

of free surface flows using various numerical analysis methods. Although inunda-tion areas and forces acting on structures can be modeled well by such simulations,the deformation and destruction of structures and the way in which water entersdamaged structures are yet to be well understood. Clarifying these mechanismswill be of great value to disaster mitigation design. To this end, Fluid-StructureInteraction (FSI) analysis involving free surface flows is required.

The Finite Element Method (FEM) has been widely used for FSI analyses [Bazilevs,Calo, Hughes, and Zhang (2008); Takizawa and Tezduyar (2011)]; however, inmany of these applications, such as the vibration of airfoils, no free surfaces werepresent. In mesh-based methods including FEM, the fluid dynamics equationshave a Eulerian basis, which means they have difficulty in dealing with a freesurface. The level set method, the Volume Of Fluid (VOF) method, and the Ar-bitrary Lagrangian-Eulerian (ALE) method have been developed and applied forcomputing free surface flows and FSI problems involving free surfaces. Recently,the Particle Finite Element Method (PFEM) was developed [Oñate, Idelsohn, Pin,and Aubry (2004)] and used to solve challenging problems [Idelsohn, Marti, Li-mache, and Oñate (2008); Ryzhakov, Rossi, Idelsohn, and Oñate (2010); Oñate,Celigueta, Idelsohn, Salazar, and Suarez (2011)]. This is an FEM-based full La-grangian description of a fluid; therefore, it readily expresses free surfaces as thefluid domain boundaries. However, PFEM has a relatively high computational costdue to remeshing.

Mesh-free particle methods, such as the Smoothed Particle Hydrodynamics (SPH)method [Lucy (1977)] and the Moving Particle Simulation (MPS) method [Koshizukaand Oka (1996)], have also attracted attention. They discretize strong form equa-tions and do not need node connectivity information. Based on a Lagrangian de-scription of the fluid, they can deal with free surfaces easily and do not have thecomputational cost associated with mesh generation. Mesh-free particle methodshave been applied to the dynamic analysis of solids and used to solve FSI problems[Antoci, Gallati, and Sibilla (2007); Rafiee and Thiagarajan (2009)]. However, incontrast with the FEM automatically satisfying the Neumann boundary conditionon the edge of a structure, mesh-free particle methods address the condition explic-itly due to strong formulations.

In recent years, a coupling approach using the FEM in structures and mesh-free par-ticle methods in fluids has been investigated for use in FSI analysis. The FEM/SPHcoupling method [Lu, Wang, and Chong (2005)] was originally developed to com-pute blast and penetration analyses and was later applied to FSI analysis [Yang,Jones, and McCue (2012)].

We developed the MPS-FE method [Mitsume, Yoshimura, Murotani, and Yamada(2014)] which adopts the MPS method for fluid computation involving free surfaces

Improved MPS-FE Fluid-Structure Interaction Coupled Method 231

and the FEM for structure computation. The method combines the advantages ofboth methods and achieves efficiency and robustness as a result. However, theconventional MPS-FE method, in which MPS wall boundary particles and finiteelements are overlapped in order to exchange information on fluid-structure in-terfaces, has difficulty in dealing with complex shaped fluid-structure boundaries,because the wall particles have to be set as uniform grids for accurate execution ofthe MPS. The result is cumbersome interpolation for the exchange of physical val-ues and node-particle correspondence relation data for the interpolation are needed.Thus, the conventional MPS-FE method lacks versatility and reduces the softwaremodularity.

In the present study, we improve the conventional MPS-FE method by using theMPS polygon wall boundary model [Harada, Koshizuka, and Shimazaki (2008);Yamada, Sakai, Mizutani, Koshizuka, Oochi, and Murozono (2011)] instead ofthe MPS wall particle model. Since the MPS polygon wall boundary model canexpress wall boundaries as plane polygons, the interface in the fluid computationnow corresponds to that in the structure computation. This improvement overcomesthe problems in the conventional method and improves versatility.

The outline of the present paper is as follows. The discretizations of the governingequations for the MPS method and the FEM are presented in Section 2 and Section5, respectively. In Section 3, we describe the computation of the pressure on thepolygons that is necessary to apply the polygon model to the MPS-FE computation,and discuss the accuracy and applicability in Section 4. Considering the pressure onthe polygons, the formulation and algorithm of the improved MPS-FE is presentedin Section 6. In order to verify the proposed method, an FSI problem involving afree surface is solved in Section 7. Conclusions complete the paper in Section 8.

2 MPS formulation for fluid dynamics

2.1 Governing equations

The Navier-Stokes equations and the continuity equation for a quasi-incompressibleNewtonian fluid in a Lagrangian reference frame are given as follows:

DvvvDt

=− 1ρ

∇p+ν f ∇2vvv+ggg (1)

1ρ

∂ρ

∂ t+∇ · vvv = 0 (2)

where ρ is the density of the fluid, vvv is the velocity vector, p is the pressure, ν f isthe kinetic viscosity, and ggg is the gravitational acceleration vector.


2.2 MPS discretization

In the MPS discretization, the differential operators acting on a particle i are evalu-ated using the neighboring particles j that are located within an effective radius re.Pi represents the set of the neighboring particles of the particle i as follows:

Pi = j | re > |xxxi j|∧ j 6= i. (3)

In this section, φi j denotes φ j− φi, where φ represents a property of the particle.The neighboring particles are weighted using the weight function of their separationfrom particle i, r = |xxxi j|. In the original MPS, the weight function is given as

w(r) =

re

r−1 (0≤ r < re)

0 (re ≤ r).(4)

To calculate the weighted average, a normalization factor termed the particle num-ber density is defined as

ni = ∑j∈Pi

w(|xxxi j|). (5)

In the MPS method the fractional step algorithm is applied for time discretization,so each time step is divided into prediction and correction steps, as follows:1© Prediction stepvvv∗− vvvn

∆t= ν f

⟨∇

2vvv⟩n

+ggg (6)

2© Correction step

vvvn+1− vvv∗

∆t=−

1ρ〈∇p〉n+1 . (7)

Here vvv∗ is the intermediate velocity. The angle brackets 〈〉 indicate discretizationby the MPS differential operators which are given as follows:

〈∇p〉i =dn0 ∑

j∈Pi

[xxxi j

|xxxi j|pi j

|xxxi j|w(|xxxi j|)

](8)

⟨∇

2vvv⟩

i =2d

λ 0n0 ∑j∈Pi

[vvvi jw(|xxxi j|)] . (9)

Here d is the number of dimensions and n0 is the initial value of the particle numberdensity given by Eq.(5). Similarly to n0,λ 0 is a quantity calculated for the initialgeometry:

λi =∑ j∈Pi |xxxi j|2w(|xxxi j|)

∑ j∈Pi w(|xxxi j|). (10)


2.3 Pressure calculation

To calculate the correction step, Eq.(7), the pressure values in the next time steppn+1 are required. For the pressure calculation, there are Semi-Implicit MPS (SI-MPS) methods [Koshizuka and Oka (1996)] in which the pressure Poisson equationis solved implicitly to determine the pressure values and an Explicit MPS (E-MPS)[Shakibaenia and Jin (2010)] method that assumes weak compressibility.

The present study employs the E-MPS method to compute the FSI with a freesurface and is validated in Section 7. Assuming the weak compressibility of fluids,the explicit calculation of the pressure can be written as

pn+1i = c2

ρ

(n∗in0 −1

)(11)

where c is a parameter set arbitrarily to satisfy the conditions of stability and in-compressibility. In addition, the E-MPS uses the following pressure gradient modelinstead of Eq.(8):

〈∇p〉i =dn0 ∑

j∈P

[xxxi j

|xxxi j|p j + pi

|xxxi j|w(|xxxi j|)

]. (12)

3 MPS polygon wall boundary model

In general, MPS computations use wall particles to model wall boundaries as il-lustrated in Fig. 1. However, since the particles have to be set in uniform grids tocorrectly calculate the particle number density defined by Eq.(5), the wall particlemodel ineffectively simulates complex shaped boundaries. Harada et al. proposedthe polygon wall boundary model [Harada, Koshizuka, and Shimazaki (2008)]which can express wall boundaries as plane polygons. By applying the polygonwall boundary model to an SI-MPS computation, they showed that the number ofmatrix elements in the pressure Poisson calculation and the computation time arereduced. A polygon wall boundary model applying the E-MPS has also been pro-posed [Yamada, Sakai, Mizutani, Koshizuka, Oochi, and Murozono (2011)]. Thedifferences between these two methods stem from the character of the pressure gra-dient term. In the present study, we use Yamada’s polygon wall boundary modelbecause it has higher stability than Harada’s.

3.1 Wall weight function

To implement the polygon wall with no particles, the interpolation of the contribut-ing part (green area at the right of Fig. 1) within the particle number density compu-tation is required. The wall weight function is defined by the sum of the virtual wall


Figure 1: Wall particle model and polygon wall model

particles’ weights created inside the wall. Using the distance between the particleand the wall |xxxib|, the wall weight function z(|xxxib|) is defined as

z(|xxxib|) = ∑j∈wall

w(|xxxi j|). (13)

Within the neighboring particles j ∈ Pi of particle i, we define the fluid particles asj ∈ particle and the virtual wall particles as j ∈ wall. The particle number densityEq.(5) can be divided into the contributions of the fluid particles and the wall weightfunction as follows:

ni = ∑j∈particle

w(|xxxi j|)+ ∑j∈wall

w(|xxxi j|)

= ∑j∈particle

w(|xxxi j|)+ z(|xxxib|). (14)

In the actual computation, the wall weight function is determined by the linear inter-polation of values at the discrete distance d0,d1, · · · ,dn(= re), which are calculatedin advance of fluid computation.

3.2 Viscosity term

The viscosity term discretized by the Laplacian model (9) is divided into fluid par-ticle and polygon wall contributions:⟨∇

2vvv⟩

i =⟨∇

2vvv⟩particle

i +⟨∇

2vvv⟩wall

i . (15)

Assuming that the velocity of the wall b is a constant value vvvb within the effectiveradius of the particle i, the contributing region of the polygon wall can be deformed


as follows:⟨∇

2vvv⟩wall

i =2d

λ 0n0 vvvib ∑j∈wall

w(|xxxi j|) (16)

=2d

λ 0n0 vvvib z(|xxxib|). (17)

Substituting the term into Eq.(15), the equation for the viscosity in the polygon wallmodel is derived as follows:⟨∇

2vvv⟩

i =2d

λ 0n0 ∑j∈particle

[vvvi jw(|xxxi j|)]+2d

λ 0n0 vvvib z(|xxxib|). (18)

3.3 Pressure gradient term

As is the case with the viscosity term, the pressure gradient term is divided into thefluid particle and polygon wall contributions as follows:

〈∇p〉i = 〈∇p〉particlei + 〈∇p〉wall

i . (19)

As mentioned, the present study adopts Yamada’s pressure gradient model [Ya-mada, Sakai, Mizutani, Koshizuka, Oochi, and Murozono (2011)]:

〈∇p〉walli =

dn0

xxxib

|xxxib|pi

|xxxib|z(|xxxib|) (20)

〈∇p〉i =dn0 ∑

j∈particle

[xxxi j

|xxxi j|p j + pi

|xxxi j|w(|xxxi j|)

]+

dn0

xxxib

|xxxib|pi

|xxxib|z(|xxxib|). (21)

3.4 Pressure on the polygon wall

The computation of the pressure on a polygon is described briefly in Yamada’sreport, but is not clearly defined. Therefore, we formulate it here.

Let Wb be the set of fluid particles that are influenced by the presence of the poly-gon wall b. The pressure on the polygon wall surface is the force exerted by thefluid particles per unit area. As shown in Fig.2, the force on the polygon wall b ex-erted by fluid particles fff b is the reaction to the sum of the pressure gradient forcesfff b j exerted by every particle j belonging to the set Wb as follows:

fff b =− ∑j∈Wb

fff jb (22)

fff jb =−mi

ρ〈∇p〉wall

j (23)


where mi is the fluid density and V is the volume occupied by a fluid particle.Assuming that all fluid particle radii are identical and the initial particle spacingis l0, the mass of each fluid particle can be calculated as mi = ρ(l0)d . Thus, thepressure on the polygon wall can be computed as follows:

pb =

∑j∈Wb

(l0)d 〈∇p〉wallj

sb(24)

where sb is the area of the polygon b.

Figure 2: The force on the polygon wall exerted by the fluid particles

4 Hydrostatic pressure calculation using MPS polygon wall boundary model

In order to verify the accuracy of the polygon wall boundary model with E-MPSand the pressure computation described by Eq.(24), we analyzed a hydrostatic pres-sure problem in a rectangular vessel and compared the result with the theoreticalsolution. The theoretical hydrostatic pressure p is given by the following equation:

p = ρ|ggg|h (25)

where h is the depth from the static water surface.

The initial configuration of the hydrostatic pressure problem is shown in Fig .3.The rectangular vessel’s depth is 0.1[m] and the width is 0.04[m]. We performedsimulations with several numbers of polygons for the right hand wall and computedthe pressures on the wall surfaces using Eq.(24). In E-MPS, weak compressibilitycauses vertical vibrations of the fluid surface. To reach the static state as soon


0.04

0.1

[m]

Figure 3: Hydrostatic pressureproblem: initial configuration

Table 1: Hydrostatic pressure problem: calcula-tion parameters

Time step width 5.0×10−5[s]Number of particles 4000Particle spacing 1.0×10−3[m]Effective radius 2.9l 0[m]Density 1.0×103[kg/m3]Kinetic viscosity 1.0×10−4[m2/s]Gravitational acceleration 9.8[m/s2]

as possible, we chose the highest value of viscosity that would not destabilize thecomputations. Figure 4, Fig. 5, Fig. 6, and Fig. 7 show the average and standardvariation of the pressure values from 40000th to 41000th steps into the simulation,with walls comprising 10, 50, 100, and 150 polygons, respectively.

Table 2: Ratio of polygon length to particle diameter in each division

Number of polygons Polygon length : particle diameter

10 10 : 150 5 : 1

100 1 : 1150 2/3 : 1

The ratio of the polygon length to the particle diameter is shown in Table 2. Theaverage pressure values agree well with the theoretical values when the polygonlength is large compared to the particle diameter, as shown in Fig. 4 and Fig. 5.In addition, since the pressure on the polygon is equivalent to the spatial averageof the computed pressure, the pressure dispersion is reduced in the case of longerpolygons. Several polygons have low pressure values in the cases where the poly-gon length is the same or shorter than the particle diameter, shown by Fig. 6 andFig. 7. This occurs when polygons are not interacting with any particles, i.e.,Wb = /0 in Eq.(24). In this case, a non-smooth pressure distribution on the wallboundary results. In FSI simulations in particular, this in turn may cause unreason-


0

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0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Pre

ssur

e on

wal

l sur

face

[N/m

2 ]

Depth [m]

Yamada polygon model (10 polygons)Theoretical value

Figure 4: Pressure on the right wall(10 polygons)

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ably high-order vibration modes in the structure and destabilize the computation;therefore, the polygon length should be set to more than twice the particle diameterfor stability of computation.

5 FE formulation for structure dynamics

5.1 Governing equations for finite deformation

The virtual work equation of the total Lagrangian formulation with geometric non-linearity is given as follows:∫

Ωs

ρ∂ 2uuu∂ t2 ·δuuudΩ +

∫Ωs

SSS : δEEEdΩ −∫

Ωs

ρggg ·δuuudΩ −∫

Γs

ttt ·δuuudΓ = 0 (26)

where Ωs is the structure dynamics domain with boundary Γs, SSS is the second Piola-Kirchhoff stress tensor, EEE is the Green-Lagrange strain tensor, ρ is the density of


the structure, and ttt is the boundary traction in the reference configuration. EEE andSSS in Eq.(26) are expressed using the deformation gradient tensor FFF = ∂uuu

∂XXX + III asfollows:

EEE =12(FFFT FFF− III) (27)

SSS =CCC : EEE (28)

where III is the second-order identity tensor andCCC is the fourth-order elastic modulustensor. In the present study, the Saint Venant-Kirchhoff model, which represents thelinear relationship between the second Piola-Kirchhoff stress tensor and the Green-Lagrange strain tensor, was used. In this case,CCC is expressed using the fourth-orderidentity tensor III and Lamé constants λ and µ as follows:

CCC= λ III⊗ III +2µIII. (29)

5.2 Solution of non-linear problems

Because of the non-linear term in Eq.(26), non-linear solvers such as the Newton-Raphson method are required. Using the Newton-Raphson method, the discretizedform of Eq.(26),

ΨΨΨ≡ fff −RRR(uuu)−MMMuuu = 0, (30)

is solved iteratively as follows:

KKKiT duuui

n+1 = ΨΨΨin+1 (31)

uuui+1n+1 = uuui

n+1 +duuuin+1. (32)

Here n is the time step, i is the iteration counter, fff is the external force vector, RRR(uuu)is the internal work vector, and MMM and KKKT are the global mass matrix and the globaltangent matrix, respectively, which are assembled from each element matrix.

5.3 Time integration scheme

The time integration scheme used in the present study is Newmark’s β method :

uuun+1 = uuun +∆tuuun +(12−β )∆t2uuun +β∆t2uuun+1 (33)

uuun+1 = uuun +(1− γ)∆tuuun + γ∆tuuun+1. (34)

In this study, we use the coefficient values β = 0.3025 and γ = 0.6, which add nu-merical damping. This kind of damping has been frequently used as a stabilizationtechnique for FSI computation [Ishihara and Yoshimura (2005); Bazilevs, Calo,Hughes, and Zhang (2008)].


6 Improved MPS-FE fluid-structure interaction method

6.1 Fluid-structure interaction model

MPS-FE Improved MPS-FE

fluid particle

MPS polygon wallfinite element

fluid particle

wall particle finite element

Figure 8: Conventional and improved MPS-FE

In the present study, the force on the polygon wall b exerted by the particles jis described by fff jb. In addition, the point on the polygon surface from which aline perpendicular to the surface can extend to the particle is termed the effectiveposition of the particle. In d-dimensional computations, an effective position on apolygon wall can be expressed as a (d−1)-dimensional vector xxxΓ using an arbitrary(d− 1)-dimensional basis. Furthermore, there are shape functions in the normal-ized natural coordinate ξξξ expressed as NΓ

i (ξξξ )(i = 1,2, · · ·nΓnode) on the surfaces of

a d-dimensional solid element, where nΓnode is the number of nodes on the element

surface. The effective position xxxΓj of particle j expressed in the normalized natural

coordinate is ξξξ j. The point loads, fff particlejb , which particles j ∈Wb exert on the

polygon b are distributed as equivalent nodal forces fff nodei on the node i using the

shape function NΓi (ξξξ ) over the element surface as follows:

fff nodei = ∑

j∈Wb

NΓi (ξξξ j) fff particle

jb . (35)

The pressure on the polygon described by Eq.(24) means the pressure values areconstant. To replace Eq.(35) which expresses the fluid force as point loads, theequivalent nodal forces representing the pressure distribution p(xxx) are expressed asfollows:

fff nodei =

∫Γ

N(xxx)p(xxx)dxxx (36)

=∫

Γ

N(ξξξ )p(ξξξ )det JJJdξξξ (37)


where JJJ is the Jacobian matrix. If the pressure distribution on the polygon b iscomputed using Eq.(24), the pressure values on the polygon are constant, pb, anddet JJJ is equivalent to the area of the polygon sb. Hence, Eq.(37) can be rewritten asfollows:

fff nodei = pbsb

∫Γ

NΓi (ξξξ )dξξξ . (38)

In this study, we used Eq.(38) for the computation of the problem presented inSection 7.

6.2 Partitioned coupling scheme

In the present study, the finite deformation FEM and the E-MPS method were usedfor FSI simulations. In this case, the fluid and structure computation time in atime step are quite different because the structure problem is solved implicitly andinvolves nonlinear iterations, whereas the fluid problem is solved explicitly. Addi-tionally, the E-MPS computation requires a finer temporal resolution than the FEM.Therefore, the present study adopts a conventional serial staggered (CSS) scheme[Farhat and Lesoinne (2000)] as an FSI coupling strategy, which applies fluid com-putation to subcycling steps in order to set different values for the fluid time step ∆tsand the structure time step ∆t f . Furthermore, the pressures exerted on the structureare averaged over k = ∆ts/∆t f fluid subcycles in order to suppress the nonphysicalpressure oscillation in the MPS.

The procedure at the nth time step is summarized below.

1© Determine the predicted values of displacement uuun+1 and velocity ˆuuun+1 fromthe previous values.

2© Perform the MPS calculation k times and determine the averaged pressuresexerted by the MPS wall boundary particles p as follows:

p =1k

k

∑i=1

pn+ ik. (39)

3© Convert the obtained pressures p into the nodal equivalent forces fff n+1 givenby Eq.(38).

4© Perform the FEM calculation to obtain the updated displacements uuun+1 andvelocities uuun+1.


fluid analysis

structure analysis

p f

u .

n+1 ,( )u n+1

→ n+1 ①

②

③

④

tn tn+1

Φ fluid n Φ

fluid n+1

Φ structure n+1 Φ

structure n

timetn+ k i tn+

k i -1

p n+ k i

averaging

p n+ k 1 p n+ 1

^ ^

Figure 9: Presented weak coupling scheme based on CSS

7 Verification of the improved MPS-FE method

In order to verify the improved MPS-FE method, we solved the dam break prob-lem with an elastic obstacle and compared the results with those obtained by othermethods.

7.1 Simulation setup

The initial configuration of the dam break problem with an elastic obstacle is illus-trated in Fig. 10. This problem was initially solved by [Walhorn, Kölke, Hübner,

0.292

0.2

92

0.146

0.012

0.0

8

0.584

Figure 10: Verification problem: initial configuration (units: m)


and Dinkler (2005)] using a space-time FEM and has since been investigated inother works [Idelsohn, Marti, Limache, and Oñate (2008); Ryzhakov, Rossi, Idel-sohn, and Oñate (2010); Rafiee and Thiagarajan (2009); Kassiotis, Ibrahimbegovic,and Matthies (2010)]. Calculation parameters are given in Table 3 and Table 4.

Table 3: Verification problem: fluid parameters

Time step width 1.0×10−6[s]Number of particles 10658Particle spacing 2.0×10−3[m]Effective radius 5.8×10−3[m]Density 1.0×103[kg/m3]Kinetic viscosity 1.0×10−6[m2/s]Gravitational acceleration 10.0[m/s2]

Table 4: Verification problem: structure parameters

Time step width 1.0×10−4[s]Number of elements 120Young’s modulus 1.0×106[kg/s2m]Poisson’s ratio 0.0Density 2.5×10−3[kg/m3]Gravitational acceleration 10.0[m/s2]

7.2 Simulation result and comparison with other methods

Figure 11 shows the displacement of the upper-left corner of the elastic obstacleat each time step. In the figure, the result of the improved MPS-FE method iscompared with that of the conventional MPS-FE method [Mitsume, Yoshimura,Murotani, and Yamada (2014)] and the particle FEM (PFEM) [Ryzhakov, Rossi,Idelsohn, and Oñate (2010)]. Whereas the computation of the conventional MPS-FE method in the previous study did not use any numerical damping for stabiliza-tion, the conventional MPS-FE computation shown here uses the same dampingconditions as the improved MPS-FE method as described in Section 5.3. Snap-shots of the simulation using the proposed method are shown in Fig. 12.

As shown in Fig. 11, the improved MPS-FE result is in good agreement with thePFEM result. Although the conventional MPS-FE result is also in agreement with


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lect

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[m]

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Improved MPS-FEMPS-FE

P-FEM: Ryzhakov et al.

Figure 11: Verification problem: comparison with other methods

0.24[s] 0.68[s] 1.35[s]

Figure 12: Verification problem: visualization of results


that of the PFEM to some degree, unnatural vibration after the initial deflection anda phase shift after 0.8[s] are observed. As mentioned previously, in the conven-tional MPS-FE approach, which uses the pressure values on the wall particles toexchange physical values, the forces on the fluid particles exerted by the structureare not consistent with the force on the structure computed by interpolation usingthe pressure on the wall particles. Since this inconsistency causes the fluid particlesto exhibit unstable behavior near the fluid-structure interface, some fluid particlesbecome too close or too far from the wall particles. The result is incorrect freesurface determination and unreasonably high pressure values near the interface,leading to unnatural vibrations. The phase shift after 0.8[s] in the conventionalMPS-FE method also comes from the inconsistency in exchanging physical values,which causes excessive decays in the energy. In converting the pressure on the wallparticles to the equivalent nodal forces, energy is not conserved. In contrast, theimproved MPS-FE method overcomes the problems in the transmission of forcefrom the fluid to the structure by computing the force on the polygon wall directlyusing Eq.(24).

8 Conclusions

In the present paper, the improved MPS-FE method was proposed in which theMPS polygon wall boundary model is applied instead of using MPS wall particles,as were adopted in the conventional MPS-FE method. The computation methodused to evaluate the pressure on the polygon wall was presented and the FSI modelfor the improved MPS-FE method was formulated. To verify the proposed method,the dam break problem with an elastic obstacle was solved. Comparing the result ofthe proposed method with that of the conventional method and PFEM indicated thatthe improved MPS-FE has adequate accuracy and higher stability and versatility.

Acknowledgement: The present paper was supported by the JST CREST project"Development of a Numerical Library Based on Hierarchical Domain Decomposi-tion for Post Petascale Simulation".

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